The Characteristic of the Minimal Ideals and the Minimal Generalized Ideals in Rings

Abstract

In this paper, we prove that the characteristic of a minimal ideal and a minimal generalized ideal, which is meant to be one of minimal left ideal, minimal right ideal, bi-ideal, quasi-ideal, and -ideal in a ring, is either zero or a prime number . When the characteristic is zero, then the minimal ideal (minimal generalized ideal) as additive group is torsion-free, and when the characteristic is , then every element of its additive group has order . Furthermore, we give some properties for minimal ideals and for generalized ideals which depend on their characteristics.

1. Introduction

Throughout this paper by ring, we shall mean an associative ring which does not necessarily has an identity element. The motivation for this paper steams from a proposition in [1] which states that the characteristic of a simple ring is either zero or a prime number. This result for the characteristic of a simple ring, which is the analogue of a minimal ideal of ring itself, is extended for the minimal ideals and the minimal generalized ideals. By generalized ideals, we shall mean the left ideals, the right ideals, the quasi-ideals [2], the bi-ideals [3], and -ideals for non-negative integers [4].

Firstly, we will prove this theorem: the characteristic of a minimal ideal (generalized ideal) in a ring is either zero or a prime number . If the , then the minimal ideal (generalized ideal) regarded as an additive group is torsion-free. If the , when is prime number, then every element of additive group is of order . Furthermore, by this theorem, we provide some results about minimal ideals and minimal generalized ideals. Thus, among others, we get that a zero ring (a ring with zero multiplication) whose characteristic is not a prime number cannot be a minimal ideal (minimal generalized ideal) of any ring. By this corollary, we obtain infinitely many counterexamples of an open problem raised in [2] and solved for the first time with only a counterexample in [5].

For the minimal left ideal, the minimal right ideal and the minimal quasi-ideal have their characteristic zero, and it is proved that for every prime number , their additive groups are -divisible. Moreover, for the minimal quasi-ideal of a ring which has the intersection property (this means that it is the intersection of a left ideal and a right ideal) and has the characteristic zero (characteristic ), we prove that if is -equivalent -equivalent, -equivalent) with the element , then the minimal quasi-ideal has the characteristic zero (characteristic ).

Finally, at the end of this paper, a simplification of the Proposition 5 [6] formulation is given and consequently a shorter proof for it. The original proof in [6] was a bit more complicated, and giving an easier proof of it was another motivation for us.

In this paper, two open problems arise in the last section.

This work is the extension of effort to give some properties for minimal ideals and for generalized ideals in rings which depend on their characteristics.

2. Preliminaries

We present some notions and some auxiliary results that will be used throughout the paper.

Definition 1. (see [7]). An abelian group is called -divisible for a prime , and if for every , there exists such that .

Definition 2. (see [8]). An abelian group is said to be torsion-free group if no element other than the identity is of finite order.
As generalized ideals in a ring, we shall consider the left ideals, the right ideals, the quasi-ideals, the bi-ideals, and the -ideals where are every non-negative integers, whose definitions are as follows ( is suppressed if ):

Definition 3. (see [2]). A quasi-ideals of a ring is called an additive subgroups of if .

Definition 4. (see [3]). A subring of a ring is called a bi-ideal if .

Definition 5. (see [4]). For every non-negative integers , a subring of the ring is called a -ideal if .
Let be an element of a ring . The intersection of all ideals (generalized ideals) of which contains the element is called the principal ideal (the principal generalized ideal) of generated by . The principal ideal (left ideal, right ideal, quasi-ideal) is denoted by (, , , respectively). It is easy to see that for every element of a ring , we have

Proposition 1 (see [1]). Let be a nonzero element of a prime ring and suppose that there is a least positive integer such that . Then, has characteristic , and is a prime number.

In [6], Green’s relations were introduced and studied, which are the analogues of Green’s relations , , in plain semigroups [9]. So Green’s relations in a ring are defined by

It turns out that are equivalence relations. The respective equivalence classes of are denoted by , , and , respectively.

In [6], it is proved that commute, that is, , and also that is an equivalence relation. So, we have the forth Green’s relation on , which is . The equivalence class containing is denoted by .

A quasi-ideal in a ring is said to have the intersection property if it is the intersection of a left ideal and a right ideal.

Theorem 1 (see [6]). Let be two elements of the ring such that . The principal quasi-ideal is minimal and has the intersection property if and only if the same hold for .

In [6], it is also introduced and studied the relation “to generate the same principal quasi-ideal,” that is,

It is easy to see that the relation is an equivalence relation and . In [6], differently from semigroups, it is shown that the inclusion is strict, and the following proposition is proved.

Proposition 2 (Proposition 5 [6]). If a minimal quasi-ideal in a ring is a subring, whose characteristic is not a prime number, then is an -class.

In [10], the authors have proved that which of Green’s relations and in rings preserve the minimality of quasi-ideal. By this, it was shown the structure of the classes generated by the above relations which have a minimal quasi-ideal.

3. Main Results

Since every simple ring is prime, then by the Proposition 1, it follows that the characteristic of a simple ring is either zero or a prime number. The same is true for every minimal ideal (generalized ideal) as the following theorem shows, which gives also a further property regarding its characteristic.

Theorem 2. The characteristic of a minimal ideal (generalized ideal) in a ring is either zero or a prime number . If the , then the minimal ideal (generalized ideal), regarded as an additive group, is torsion-free. If the , then every element of the additive group is of order .

Proof. We will give the proof only for the minimal ideal, and the proof for others is similarly.
We will consider two possible cases: First case: . In this case, for every positive integer , there exists an element such that . For every element , we have Then, there exists an integer and the elements of such that If there is a positive integer , such that , then we shall getwhich is a contradiction. Therefore, the order of any nonzero element of additive group is infinite, and consequently, the group is torsion-free. Second case: , where is a positive integer not smaller than two. So, for every element of , we have , and for every positive integer such that , there exists an element of such that . Since , then again the equality (5) holds true. By this equality, it follows that for every element of , we have . Thus, the order of every element of the additive group is . By the Cauchy Theorem, for every prime divisor of , there exists an element of additive group which has its order . Since , then we find the integers and the elements of the ring such that By this equality, we have If there is a positive integer such that , then we would have . Since , then there exist an integer and the elements of the ring such that By this equality, we get Thus, we have the equality which is a contradiction Therefore, , that is , where is a prime number and the order of any element of is Since every simple ring is a minimal ideal of , then bythe Theorem 2, we have the following two corollaries (the first part of the first corollary derives from Proposition 2):

Corollary 1. (1)If is a simple ring, then its characteristic is either zero or a prime number(2)If , then the additive group is a torsion-free group(3)If , prime, then every element of additive group has order

Corollary 2. If is a simple ring, then we have(1) if and only if for every minimal generalized ideal in (2), prime number, if and only if for every minimal generalized ideal in By Theorem 2, we get immediately the following corollary:

Corollary 3. If the characteristic of a ring is not a prime number, then there is no ring such that is a minimal ideal or minimal generalized ideal of .

By the above corollary, it follows that the ring of integers modules , where is not a prime number, , the ring of matrices with elements in , and also the ring of polynomials with coefficients in can not be minimal ideal or minimal generalized ideal of some ring.

As a particular case of Corollary 3, we have that any ring with zero multiplication (zero ring) which have its characteristic not a prime number cannot be a minimal ideal or a minimal generalized ideal of some ring.

For the quasi-ideal, regarded as a generalized ideal, with zero multiplication [2], L. Márki has raised this problem: given a zero ring does, there exist a ring such that is a minimal quasi-ideal of .

This open problem is resolved negatively with one counterexample in [5]. Here, we find infinity many zero rings which cannot be quasi-ideals of some ring. For example, let we consider the additive groups , for every not-prime number , the additive groups of rings of matrices with elements in , and the additive groups of polynomials with coefficients in . They can be made into requested zero rings, by defining their multiplication by , where 0 is the identity of their addition.

For the minimal left ideal, the minimal right ideal, and the quasi-ideal, the -ideal, and -ideals, , a part of Theorem 2 is sharpen by this.

Theorem 3. If the characteristic of a minimal left ideal (right ideal, quasi-ideal, -ideal, , -ideals, ) of a ring is zero, then this minimal left ideal (right ideal, quasi-ideal, -ideal, , -ideals, ) considered as an additive group is divisible for every prime .

Proof. We will give the proof only for minimal left ideal and minimal quasi-ideal: the rest of proof is similarly.
Let be a minimal left ideal in a ring such that the . By Theorem 2, we have that is a torsion-free group. Hence, for every prime and for every element , we have . Since is minimal andthen we have that . Then, there exist an integer and an element such that . By this equality, we have . The integer is not zero; therefore, by Theorem 2, we have . Since is minimal andthen we get . So, there exist an integer and the element such thatBy this equality, if we denote , then we have , where , and consequently, the minimal left ideal as an additive group is -divisible for every prime .
Now, we give the proof for a minimal quasi-ideal in the ring . By Theorem 2, for every prime and every element of , we have . Since is minimal quasi-ideal andthen we have . Then, there exist an integer and two elements of such thatBy these equalities, we findThe integer is not zero, and by Theorem 2, we have . Since is minimal quasi-ideal andthen we have . Then, there exist the integers and the elements of such thatBy these equalities, we find , where and . ButThus, , where , and consequently, the minimal quasi-ideal as an additive group is -divisible for every prime .
For the minimal generalized ideals in which there are not considered in Theorem 3, the following open problem arise:

Problem 1. Does the assertion of Theorem 3 hold true for minimal ideal, in particular for simple ring, for minimal bi-ideal, and for minimal -ideal where or ?
It is shown in [1] that there exist simple rings which do not have an identity element. It follows that there are minimal ideals in rings which do not have an identity element. On the other hand, also in [1], it is proved that every simple right Noetherian ring of characteristic zero has an identity element. Indicated by this proposition, the following open problem arises:

Problem 2. Do minimal ideals (minimal left ideals, minimal right ideals) of a right Noetherian ring of characteristic zero have identity elements?
Now, we will show that the characteristic of a minimal quasi-ideal which has the intersection property will be “transferred” from each of Green’s relations and in rings. Precisely, we have the following theorem:

Theorem 4. Let be two elements of a ring such that . Then, the principal quasi-ideal is minimal of characteristic zero (characteristic ,-prime) and has the intersection property if and only if the same holds for .

Proof. In view of the duality between and and the definition of , it suffices to prove the theorem only for the relation .
Since , then by Theorem 1, the principal quasi-ideal is minimal, and it has the intersection property if and only if the same holds for . So, it remains to show that if has the characteristic zero (characteristic , -prime), then has the characteristic zero (characteristic , -prime).
Since , then there exist an integer and an element such that . Therefore, by the Proposition 1 [6], the mappingis a bijection. Furthermore, for every two elements of , we haveThis equalities show that is an additive groups isomorphism. So, the quasi-ideals as additive groups of a ring are isomorphic groups. It now follows that the minimal quasi-ideals have the same characteristic.
By Theorem 2, the characteristic of a quasi-ideal in a ring is either zero or a prime number, and therefore, the proposition which we will prove below is equivalent with Proposition 2, but the proof that we give here is a little bit simpler and shorter than the proof of Proposition 2 (Proposition 5 [6]).

Proposition 3. If a minimal quasi-ideal of a ring has its characteristic zero, then is a -class.

Proof. Since is a minimal quasi-ideal with , then for every prime , there exists an element such that . As is a minimal quasi-ideal, we haveThen, there exist integers and elements such thatHence,By these equalities, we getThe integer is not zero, and so by Theorem 3, we have . Sinceand is a minimal quasi-ideal, then we get . Hence, there exist elements of such that . Now, let be an arbitrary element of . Then, there exist elements such that . Since is a minimal quasi-ideal, then we have , and consequently, . Thus, and finally, we have .

4. Conclusion

In this paper, we have shown that the characteristic of minimal ideals and minimal generalized ideals in rings, which do not necessarily have an identity element, is zero or a prime number. Furthermore, we gave some properties for minimal ideals and for generalized ideals that depend on their characteristics. By using these results and Green’s relations, we have shown an additional result, that is, the characteristic of a minimal quasi-ideal that has the intersection property will be transferred from each of Green’s relations in rings. More specifically, if and are two elements of a ring such that , then the principal quasi-ideal is minimal of characteristic zero (characteristic , -prime) and has the intersection property if and only if the same holds for . The obtained results led raise the following issues:(1)What happens between the principal bi–ideal and the principal bi–ideal if in a ring ?(2)Does the assertion of Theorem 3 hold true for minimal ideal, in particular for simple ring, for minimal bi-ideal and for minimal -ideal where or ?(3)Do minimal ideals (minimal left ideals, minimal right ideals) of a right Noetherian ring of characteristic zero have identity elements?(4)Can we extend these results to other structures such as hyper-rings, ternary rings, -rings etc.?

These issues will be some the potential aims of our future work.

Data Availability

No data were used to support the findings of the study because this paper is in pure mathematics (algebra) and all results are in these fields.

Conflicts of Interest

The authors declare that they have no conflicts of interest.